Reuben:

> It would be useful to get a sense of the motivation for the direction this chapter seems to be building in, namely a category where morphisms are quantale enriched profunctors.

I tried to give that motivation in the first few lectures:

* [Lecture 55 - Chapter 4: Enriched Profunctors and Collaborative Design](https://forum.azimuthproject.org/discussion/2278/lecture-55-chapter-4-enriched-profunctors/p1)

(a general introduction to the meaning of enriched profunctors as relating 'requirements' - things you want - to 'resources' - things you have... and a tiny taste of their application to collaborative design)

* [Lecture 56 - Chapter 4: Feasibility Relations](https://forum.azimuthproject.org/discussion/2280/lecture-56-chapter-4-feasibility-relations/p1)

(showing how the definition of a \\(\textbf{Bool}\\)-enriched profunctor is natural when you think of it as describing a relation between requirements and resources)

* [Lecture 59 - Chapter 4: Cost-Enriched Profunctors](https://forum.azimuthproject.org/discussion/2285/lecture-59-chapter-4-cost-enriched-profunctors/p1)

(a similar example, but now instead of just either being able to get from some resources to some requirements, you have to pay a certain _cost_ to get from the resources to the requirements)

In most of my examples so far I've found it easiest to talk about the process of getting this from that in terms of _travelling between cities_, because it's such a beautiful visual metaphor. But we could equally well be talking about _turning any resource into any product_.

There's a lot more motivation in [the book](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf), so please read that too!

In this particular chapter, Fong and Spivak go through the business of building a category where morphisms are quantale-enriched profunctors very rapidly, and spend most of their time drawing pictures of examples. So, I decided to spend more time on the details they skim over.

Once we get this category - which we'll have as soon as people solve the puzzles in this lecture! - I'll try to use it for something.

The main paper on this subject is:

* Andrea Censi, [A mathematical theory of co-design](https://arxiv.org/abs/1512.08055).

If you have more precise questions, please ask!